Ehrenfeucht theories with non-dense powerful digraphs*

Abstract

All known Ehrenfeucht theories are based on powerful digraphs having the density property. In the paper, we construct examples of Ehrenfeucht theories with powerful digraphs having the non-density property, i. e., with covering elements for each vertex of digraph. The construction of required theories is realized by syntactic generic ones allowing to synthesize saturated models based on the class of finite structures with some type add-ins, closed under amalgams.